Methods and systems of fast optimization and compensation for volumetric positioning errors of rotary axes of five-axis cnc machine tools

ABSTRACT

Embodiments of the present disclosure provide a method of fast optimization and compensation for volumetric positioning errors of rotary axes of a five-axis CNC system machine tool. The method comprises: establishing a volumetric positioning error model; forming an error database containing 12 geometrical error vectors; constructing a volumetric positioning error compensation table; establishing a compensation value optimization model; completing an iterative optimization of compensation values of volumetric positioning errors; generating a volumetric positioning error compensation file for a CNC system to complete compensation for the volumetric positioning errors; and updating the error database, detecting linkage trajectories of the rotary axes, and setting a linkage trajectory positioning error threshold, and guaranteeing accuracy by iteratively implementing detection, optimization, and compensation.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of International ApplicationNo. PCT/CN2021/136930, filed on Dec. 10, 2021, which claims priority toChinese Patent Application No. 202110187158.4, filed on Feb. 18, 2021,the entire contents of each of which are hereby incorporated byreference.

TECHNICAL FIELD

The present disclosure relates to the field of accuracy compensation forfive-axis computer numerical control (CNC) machine tools, and inparticular, to a method of fast optimization and compensation forvolumetric positioning errors of rotary axes of a five-axis CNC machinetool.

BACKGROUND

A five-axis computer numerical control (CNC) machine tool is a coredevice for processing large and complex curved parts. The machiningaccuracy of the five-axis CNC machine tool directly determines thequality of machined parts. As the five-axis CNC machine tool has twomore rotary axes than the three-axis machine tool, the addition ofrotary axes brings machining advantages, but also brings more geometricerror sources, making it difficult to effectively guarantee themachining accuracy of the five-axis CNC machine tool. Therefore, the wayto improve the machining accuracy of the five-axis CNC machine tool hasbecome a research focus of scholars at home and abroad.

At present, the five-axis CNC machine tool made in China is comparableto those made in Europe and the United States in guaranteeing accuracyfor a translation axis, but a large gap exists in guaranteeing accuracyfor a rotary axis. To improve the machining accuracy, the five-axis CNCmachine tool focuses on improving the spatial accuracy of the rotaryaxis. Patent 201410352916 modeled and identified twelve geometric errorsin the rotary axis of a rotary platform type, and then used a geometricerror compensation value to correct a CNC instruction to achieve thecompensation effect. However, there are two shortcomings in thiscompensation method. The first one is an identified geometric error ofthe rotary axis, a value of which still exists in an operating error andan identification error, which cannot be directly used in thecompensation for the geometric error. The second one is a way ofcorrecting the CNC instruction, which has a good effect but is notsuitable for large and complex parts. This is because the number oflines of CNC instructions for large and complex parts is counted from100,000, making the way of correcting the CNC instruction tooinefficient. Patent 201610045130.6, based on the RTCP motion controlfunction, completed the identification of twelve geometric errors of arotary axis of a CA double pendulum head machine tool. An inspectiontooling is additionally arranged, so the geometric error identified bythis method may also be affected by identification accuracy, toolingerrors, etc., making the identified geometric error not directly used inthe compensation for spatial accuracy of the rotary axis. In summary,identification and compensation for the geometric error of the rotaryaxis proposed in existing methods have certain limitations. It isdesirable to provide a more adaptable method to improve spatialpositioning accuracy of the rotary axis.

In order to overcome the limitations of the existing methods, based onan identification result of a geometric error of a rotary axis, and incombination with a mathematical model of a volumetric positioning errorof a rotary axis and a non-dominated sorting genetic algorithm (NSGAIIalgorithm), some embodiments of the present disclosure provide a methodof fast optimization and compensation for volumetric positioning errorsof rotary axes of a five-axis CNC machine tool, which achieves a fastcompensation for spatial accuracy of rotary axes of the five-axis CNCmachine tool, and realizes good implementation effect.

SUMMARY

One of the embodiments of the present disclosure provides a method offast optimization and compensation for volumetric positioning errors ofrotary axes of a five-axis CNC machine tool. The method may comprise:(1) establishing a volumetric positioning error model of rotary axes ofthe five-axis CNC machine tool based on the geometric errors of therotary axes; (2) completing an identification of 12 geometric errors ofthe rotary axes based on instrument detection data to form an errordatabase containing the 12 geometric error vectors; obtaining theinstrument detection data using an inspection instrument, and obtainingidentified values of geometric errors by performing the identificationof the geometric errors using a relevant identification algorithm;wherein each of the 12 geometric error vectors may be an error vectorcombined by 6 geometric errors identified for each rotary axis of arotary axis A and a rotary axis

${{C:E_{1 \times 12}} = \underset{6}{\underset{︸}{\left\lbrack {{\delta_{x}(A)},\ldots,{\theta_{\gamma}(A)}} \right.}}},{\underset{6}{\underset{︸}{\left. {{\delta_{x}(C)},\ldots,{\theta_{\gamma}(C)}} \right\rbrack}};}$

each error term of the 12 geometric error vectors may be related to amotion position, and when a motion stroke of each rotary axis is dividedinto N equal parts, values of a same error term may be different atdifferent rotation angle positions, and the error database may be notedas:

${{Ebase}_{N \times 12} = \begin{bmatrix}{{\delta_{x}(A)}_{1},\ldots,{\theta_{\gamma}(A)}_{1},{\delta_{x}(C)}_{1},\ldots,{\theta_{\gamma}(C)}_{1}} \\\ldots \\{{\delta_{x}(A)}_{N},\ldots,{\theta_{\gamma}(A)}_{N},{\delta_{x}(C)}_{N},\ldots,{\theta_{\gamma}(C)}_{N}}\end{bmatrix}};$

where Ebase_(N×12) denotes a matrix of N rows and 12 columns; (3)decomposing the volumetric positioning errors of the rotary axes intolinear correlation and nonlinear correlation, and constructing avolumetric positioning error compensation table of the rotary axes bycombining with a sag error compensation function of a numerical controlsystem; (4) adding correction coefficients k and d to the error databaseand correlating the correction coefficients k and d with a linkagetrajectory positioning error model to establish a compensation valueoptimization model for the volumetric positioning errors, wherein thecompensation value optimization model is used to optimize compensationvalues for the volumetric positioning errors of the rotary axes; (5)carrying out compensation quality control on correction coefficientvectors K and D composed of the correction coefficients k and d based ona non-dominated sorting genetic algorithm (NSGAII algorithm) to realizeoptimization selection of the correction coefficient vectors, so as tocomplete an iterative optimization of the compensation values for thevolumetric positioning errors; (6) generating a volumetric positioningerror compensation file for the CNC system by using the correctioncoefficient vectors and the error database obtained in step (5), so asto complete the compensation of the volumetric positioning errors; and(7) updating corrected geometric errors to the error database, andperiodically detecting linkage trajectories of the rotary axes, andsetting a linkage trajectory positioning error threshold, therebyguaranteeing spatial accuracy of the five-axis CNC machine tool byiteratively implementing detection, optimization, and compensation.

For any structure type of machine tool, compensation of the volumetricpositioning errors of the rotary axes can be realized by adopting thefast optimization method in combination with the volumetric positioningerror model of the rotary axes and the geometric error optimizationmodel on the basis of the identified geometric errors of the geometricerrors.

Some embodiments of the present disclosure include at least thefollowing beneficial effects: (1) the method has higher operationconvenience and detection efficiency: after the all-round identificationis completed, detection and compensation of the volumetric positioningerrors of the rotary axes can be completed by only implementing theCA-axis linkage trajectories, so that the linkage detection greatlyimproves the operation convenience compared to detection of anindividual rotary axis, making the detection more efficient; (2)optimization and compensation have good adaptability: the geometricerrors are optimized and then compensated, the requirements for theidentification accuracy of the geometric errors are low, i.e., thegeometric errors with low identification accuracy can also becompensated and corrected through the optimization of the presentinvention, which reduces the requirements for the identificationaccuracy of the geometric errors, the requirements for the accuracy ofdetection tooling, etc., and expands the adaptability of the presentinvention; (3) the present invention has a high degree of automation offast optimization: compared to the conventional error correction method,which mostly repeats tests and directly averages the results of aplurality of experiments for compensation and correction, making thetraditional method inefficient in implementation, the present methodadopts an intelligent optimization algorithm, and performs refinedcorrection for the geometric errors by using the powerful processingcapability of the computer, thereby improving the quality ofcompensation, and facilitating fast compensation.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure is further illustrated in terms of exemplaryembodiments. These exemplary embodiments are described in detailaccording to the drawings. These embodiments are non-limiting exemplaryembodiments, in which like reference numerals represent similarstructures, wherein:

FIG. 1 is a flowchart illustrating an exemplary fast optimization ofvolumetric positioning errors of rotary axes of a five-axis CNC machinetool according to some embodiments of the present disclosure;

FIG. 2 is an exemplary flowchart illustrating fast optimization ofvolumetric positioning errors of rotary axes of a five-axis CNC machinetool according to some embodiments of the present disclosure;

FIG. 3 is a schematic diagram illustrating an example of a CA verticalfive-axis CNC machine tool according to some embodiments of the presentdisclosure;

FIG. 4 is a schematic diagram illustrating a composition of geometricerrors of rotary axes of a CNC machine tool according to someembodiments of the present disclosure;

FIG. 5 is a schematic diagram illustrating a principle of correctioncoefficients for geometric error terms of a CNC machine tool accordingto some embodiments of the present disclosure;

FIG. 6 is a graph illustrating detection of positioning errors of CAlinkage trajectories of a CNC machine tool according to some embodimentsof the present disclosure;

FIG. 7A is a graph illustrating comparison of positioning errors oflinkage trajectories and detected positioning errors before fastoptimization according to some embodiments of the present disclosure;

FIG. 7B is a graph illustrating comparison of positioning errors oflinkage trajectories and detected positioning errors after fastoptimization according to some embodiments of the present disclosure;

FIG. 8 is a schematic diagram illustrating an exemplary application of acompensation evaluation model according to some embodiments of thepresent disclosure; and

FIG. 9 is a schematic diagram illustrating an exemplary application of acompensation time point prediction model according to some embodimentsof the present disclosure.

DETAILED DESCRIPTION

To more clearly illustrate the technical solutions related to theembodiments of the present disclosure, a brief introduction of thedrawings referred to the description of the embodiments is providedbelow. Obviously, the drawings described below are only some examples orembodiments of the present disclosure. Those having ordinary skills inthe art, without further creative efforts, may apply the presentdisclosure to other similar scenarios according to these drawings.Unless obviously obtained from the context or the context illustratesotherwise, the same numeral in the drawings refers to the same structureor operation.

It should be understood that “system”, “device”, “unit” and/or “module”as used herein is a manner used to distinguish different components,elements, parts, sections, or assemblies at different levels. However,if other words serve the same purpose, the words may be replaced byother expressions.

As shown in the present disclosure and claims, the words “one”, “a”, “akind” and/or “the” are not especially singular but may include theplural unless the context expressly suggests otherwise. In general, theterms “comprise,” “comprises,” “comprising,” “include,” “includes,”and/or “including,” merely prompt to include operations and elementsthat have been clearly identified, and these operations and elements donot constitute an exclusive listing. The methods or devices may alsoinclude other operations or elements.

The flowcharts used in the present disclosure illustrate operations thatsystems implement according to some embodiments of the presentdisclosure. It should be understood that the previous or subsequentoperations may not be accurately implemented in order. Instead, eachstep may be processed in reverse order or simultaneously. Meanwhile,other operations may also be added to these processes, or a certain stepor several steps may be removed from these processes.

FIG. 1 is a flowchart illustrating an exemplary fast optimization ofvolumetric positioning errors of rotary axes of a five-axis CNC machinetool according to some embodiments of the present disclosure. In someembodiments, a process 100 may be performed by a CNC system of thefive-axis CNC machine tool. For example, the process 100 may be storedin a storage device of the CNC system in a form of a program or aninstruction, and the process 100 may be implemented when a processingdevice of the CNC system executes the program or the instruction. Aschematic diagram illustrating operation of the process 100 presentedbelow is illustrative. In some embodiments, the process may be completedby one or more additional operations not described or one or moreoperations not discussed. Further, an order of the operations of theprocess 100 illustrated in FIG. 1 and described below is not limiting.

Step (1): a volumetric positioning error model of the rotary axes of thefive-axis CNC machine tool may be established based on the geometricerrors of the rotary axes.

Step (2): an identification of 12 geometric errors of the rotary axesmay be completed based on instrument detection data, and an errordatabase containing 12 geometric error vectors may be formed.

Step (3): the volumetric positioning errors may be decomposed intolinear correlation and nonlinear correlation, and a volumetricpositioning error compensation table of the rotary axes may beconstructed by combining with a sag error compensation function of theCNC system.

Step (4): correction coefficients k and d may be added to the errordatabase, and correlated with a linkage trajectory positioning errormodel to establish a compensation value optimization model for thevolumetric positioning errors.

Step (5): compensation quality control may be performed on correctioncoefficient vector K and D composed of the correction coefficient k andd based on an NSGAII algorithm to realize optimization selection on thecorrection coefficient vectors, so as to complete an iterativeoptimization on compensation values for the volumetric positioningerrors.

Step (6): a volumetric positioning error compensation file for the CNCsystem may be generated based on the correction coefficient vectors andthe error database to implement compensation of the volumetricpositioning errors.

Step (7): corrected geometric errors may be updated to the errordatabase, linkage trajectories of the rotary axes may be periodicallydetected, a linkage trajectory positioning error threshold may be set,and spatial accuracy of the five-axis CNC machine tool may be guaranteedby cyclically implementing detection, optimization, and compensation.

Specific implementation steps of the method of fast optimization andcompensation for the volumetric positioning errors are described belowin conjunction with FIG. 2 .

Step (1): The volumetric positioning error model of the rotary axes ofthe five-axis CNC machine tool may be established based on the geometricerrors of the rotary axes.

A large gantry CA vertical five-axis CNC machine tool in FIG. 3 may betaken as an example.

Step (1.1): the geometric errors of the rotatory axes may be described.

For a rotary axis C (around a Z-axis), as shown in FIG. 4 , thegeometric errors of the rotary axis C may include six items, includingthree displacement deviation errors, i.e., a displacement error along anX-direction δ_(x)(C), a displacement error along a Y-direction δ_(y)(C),and a displacement error along a Z-direction δ_(z)(C), and three angularerrors, i.e., an angular error around an X-axis θ_(α)(C), an angularerror around a Y-axis θ_(β)(C), and an angular error around a Z-axisθ_(γ)(C). For a certain rotary axis, a generic vector form of thegeometric errors of the rotary axis may be as follows:

E(i)=[δ_(x)(i),δ_(y)(i),δ_(z)(i),θ_(α)(i),θ_(β)(i),θ_(γ)(i)]^(T),i=A,B,C;

Where A, B, and C denote the rotary axes and i denotes the rotary axis.

Step (1.2): the rotary axis may be driven to rotate by a CNCinstruction, due to an influence of the geometric errors of the rotaryaxis, an actual motion matrix of the rotary axis may be as follows:

T _(i) =R(i)·E _(i) ,i=A,B,C

Where T_(i) denotes an actual motion transformation matrix of thei-axis, R(i) denotes a theoretical motion transformation matrix of therotary i-axis, and E_(i) denotes a geometric error matrix, which may bespecifically expressed as:

${{R(A)} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos\sigma_{A}} & {{- \sin}\sigma_{A}} & 0 \\0 & {\sin\sigma_{A}} & {\cos\sigma_{A}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}},{{R(B)} = \begin{bmatrix}{\cos\sigma_{B}} & 0 & {\sin\sigma_{B}} & 0 \\0 & 1 & 0 & 0 \\{{- \sin}\sigma_{B}} & 0 & {\cos\sigma_{B}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}},$ ${{R(C)} = \begin{bmatrix}{\cos\sigma_{C}} & {{- \sin}\sigma_{C}} & 0 & 0 \\{\sin\sigma_{C}} & {\cos\sigma_{C}} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}},{E_{i} = \begin{bmatrix}1 & {- {\theta_{\gamma}(i)}} & {\theta_{\beta}(i)} & {\delta_{x}(i)} \\{\theta_{\gamma}(i)} & 1 & {\theta_{\alpha}(i)} & {\delta_{y}(i)} \\{- {\theta_{\beta}(i)}} & {\theta_{\alpha}(i)} & 1 & {\delta_{z}(i)} \\0 & 0 & 0 & 1\end{bmatrix}},{i = A},B,{C;}$

Where σ_(A), σ_(B), σ_(C) denote an running angle of each rotary axis inthe CNC instruction, respectively, (δ_(x)(i), δ_(y)(i), δ_(z)(i)) denotedisplacement errors along X, Y, Z-direction, respectively, and(θ_(α)(i), θ_(β)(i), θ_(γ)(i)) denote angular errors around X, Y, Z-axisunder a workpiece coordinate system.

Step (1.3): since the five-axis CNC machine tool has only two rotaryaxes involved in the motion, a kinematic relationship of a tool centerpoint in the case of an error when a C-axis and an A-axis are in linkageoperation may be obtained according to a topology of a CA five-axis CNCmachine tool with double pendulum heads in combination with the errormatrix and the motion matrix of the rotary axis in step (1.2) asfollows:

P _(actual) =T(C)·T(A)·P _(t);

Where P_(actual) denotes actual coordinates of the tool center pointunder the workpiece coordinate system, P_(t) denotes actual coordinatesof the tool center point under a tool coordinate system, P_(t)=[0 0 −L1]^(T), and L denotes a length of a tool.

Step (1.4): in step (1.2), (1.3), ignoring that the geometric errormatrix is a theoretical motion model, a kinematic relationship of thetool center point in an ideal case may be obtained as follows:

P _(ideal) =R(C)·R(A)·P _(t);

Where P_(ideal) denotes theoretical coordinates of the tool center pointunder the workpiece coordinate system.

Step (1.5): combined with step (1.3) and step (1.4), the volumetricpositioning error model of the rotary axes of the five-axis CNC machinetool may be obtained as follows:

P _(error) =F _(actual) −P _(ideal);

According to steps (1.2) to (1.4), an expression corresponding to thevolumetric positioning errors of the tool center point of the largegantry five-axis CNC machine tool may be obtained by substituting andsimplifying a motion variation matrix and the geometric error matrixE_(i) as follows:

$P_{error} = {\begin{bmatrix}{{{\delta_{x}(A)}\cos\sigma_{C}} - {{\delta_{y}(A)}\sin\sigma_{C}} - {L{\theta_{\beta}(A)}\cos\sigma_{A}}} \\{{\cos\sigma_{C}} - {L\theta_{\alpha}(A)\cos\sigma_{A}\sin\sigma_{C}} -} \\{{L{\theta_{\gamma}(C)}\sin\sigma_{A}\cos\sigma_{C}} - {L{\theta_{\beta}(C)}\cos\sigma_{A}} -} \\{{L{\theta_{\gamma}(A)}\sin\sigma_{A}\cos\sigma_{C}} + {\delta_{x}(C)}} \\{{\delta_{\gamma}(C)} + {{\delta_{x}(A)}\sin\sigma_{C}} + {L{\theta_{\alpha}(C)}\cos\sigma_{A}} -} \\{{L{\theta_{\gamma}(A)}\sin\sigma_{A}\sin\sigma_{C}} + {{\delta_{\gamma}(A)}\cos\sigma_{C}} +} \\{{L{\theta_{\alpha}(A)}\cos\sigma_{A}\cos\sigma_{C}} - {L{\theta_{\gamma}(C)}\sin\sigma_{A}\sin\sigma_{C}} -} \\{L{\theta_{\beta}(A)}\cos\sigma_{A}\cos\sigma_{C}} \\{{L{\theta_{\beta}(A)}\sin\sigma_{A}\sin\sigma_{C}} + {L{\theta_{\alpha}(C)}\sin\sigma_{A}\cos\sigma_{C}} +} \\{{L\theta_{\alpha}(A)\sin\sigma_{A}} + {\delta_{z}(C)} + {\delta_{z}(A)}} \\0\end{bmatrix}.}$

Step (2): the identification of 12 geometric errors of the rotary axesmay be completed based on the instrument detection data, and the errordatabase containing 12 geometric error vectors may be formed.Implementation steps are described in steps (2.1) to (2.2) below.

Step (2.1): the instrument detection data may be obtained by using adetection instrument (such as R-TEST, ball-bar, etc.), and the geometricerrors may be identified by using a relevant identification algorithm toobtain the identified geometric errors of the geometric errors.

Step (2.2): each of the 12 geometric error vectors may be an errorvector combined by 6 geometric errors identified for each rotary axiswith a rotary axis A and a rotary axis C:

$E_{1 \times 12} = {\left\lbrack {\underset{\underset{6}{︸}}{{\delta_{x}(A)},\ldots,{\theta_{\gamma}(A)}},\underset{\underset{6}{︸}}{{\delta_{x}(C)},\ldots,{\theta_{y\gamma}(C)}}} \right\rbrack.}$

Since the geometric errors of the rotary axis is interrelated with amotion position, each error term may also become a position-relatedvector throughout a motion stroke. Each error term of the 12 geometricerror vectors may be related to the motion position, i.e., when themotion stroke of the rotary axis is divided into N equal parts, valuesof a same error term may be different at different rotation anglepositions, and the error database may be noted as:

${{{Ebas}e_{N \times 12}} = \begin{bmatrix}{{\delta_{x}(A)_{1}},\ldots,{\theta_{\gamma}(A)}_{1},{\delta_{x}(C)}_{1},\ldots,{\theta_{y\gamma}(C)}_{1}} \\\ldots \\{{\delta_{x}(A)_{N}},\ldots,{\theta_{\gamma}(A)}_{N},{\delta_{x}(C)}_{N},\ldots,{\theta_{\gamma}(C)}_{N}}\end{bmatrix}};$

Where Ebase_(N×12) denotes a matrix of N rows and twelve columnsrepresenting the error database of the geometric errors of the rotaryaxes of the five-axis machine tool.

In order to realize high precision and high efficiency machining of thefive-axis CNC machine tool, it is necessary to compensate the geometricerrors of the rotary axis, thereby reducing the influence of thevolumetric positioning errors on the overall machining accuracy of themachine tool. Hereto, a volumetric positioning error compensation filemay be generated by using a sag error compensation module of the CNCsystem.

Step (3): the volumetric positioning errors may be decomposed intolinear correlation and nonlinear correlation, and a volumetricpositioning error compensation table may be constructed by combining asag error compensation function of the CNC system. Implementationoperations may be described in the following steps (3.1) to (3.3).

Step (3.1): the volumetric positioning errors may be decomposed into thelinear correlation and the nonlinear correlation by analyzing amathematical model between the volumetric positioning errors of the toolcenter point and the geometric errors of the rotary axes.

The mathematical model between the volumetric positioning errors of thetool center point and the geometric errors of the rotary axes may berepresented by:

P_(error) = P_(actual) − R_(ideal) = T(C) ⋅ T(A) ⋅ P_(t) − R(C) ⋅ R(A) ⋅ P_(t) = [R(C) ⋅ E_(C) ⋅ R(A) ⋅ E_(A) − R(C) ⋅ R(A)] ⋅ P_(t)

Taking a volumetric positioning error projected in an X-direction as anexample, the volumetric positioning errors may be decomposed into thelinear correlation and the nonlinear correlation as represented by:

P _(error_x) =L _(error_x) +N _(error_x);

Where:

L _(error_x)=δ_(x)(C);

L _(error_x)=δ_(x)(A)cos σ_(C)−δ_(y)(A)sin σ_(C) −Lθ _(β)(A)cos σ_(A)cos σ_(C) −Lθ _(α)(A)cos σ_(A) sin σ_(C) −Lθ _(γ)(C)sin σ_(A) cos σ_(C)−Lθ _(β)(C)cos σ_(A) −Lθ _(γ)(A)sin σ_(A) cos σ_(C);

Where L_(error_x) denotes a linear correlation error term, i.e., a partthat is only linearly correlated with geometric errors of a coordinateaxis; L_(error_x) denotes a nonlinear correlation error term, i.e., apart that interacts between the coordinate axis to make the positioningerrors change in a nonlinear way.

From the above, it may be seen that a large proportion of the nonlinearcorrelation error terms are not only related to error values of therotary axes at a detection point but also to relative positions of therotary axes as well as a length of the tool. Thus, it may be inferredthat the nonlinear correlation error terms are key factors affecting thevolumetric positioning errors, and also terms to focus on forcompensation.

Step (3.2): compensation of the nonlinear correlation error terms may berealized based on the sag error compensation function of the CNC system.Taking δ_(x)(A)cos σ_(C) as an example, the compensation of thenonlinear correlation error terms may be performed by the followingequation:

S(δ_(x)(A)cos σ_(C))=S(cos σ_(C))S(δ_(x)(A))

Where:

${S\left( {\cos\sigma_{c}} \right)} = {\begin{pmatrix}\left( T_{C}^{X} \right)_{h} \\\left( T_{C}^{X} \right)_{h + 1} \\\ldots \\\left( T_{C}^{X} \right)_{h + m}\end{pmatrix} = \begin{bmatrix}{\cos\sigma_{C_{1}}} & 0 & \ldots & 0 \\0 & {\cos\sigma_{C_{2}}} & \ldots & 0 \\\ldots & \ldots & \ldots & \ldots \\0 & 0 & \ldots & {\cos\sigma_{Cm}}\end{bmatrix}_{m \times m}}$${{S\left( {\delta_{x}(A)} \right)} = {\begin{pmatrix}\left( T_{A}^{X} \right)_{g} \\\left( T_{A}^{X} \right)_{g + 1} \\\ldots \\\left( T_{A}^{X} \right)_{g + m}\end{pmatrix} = \begin{bmatrix}{\delta_{x}(A)}_{1} & {\delta_{x}(A)}_{2} & \ldots & {\delta_{x}(A)}_{n} \\{\delta_{x}(A)}_{1} & {\delta_{x}(A)}_{2} & \ldots & {\delta_{x}(A)}_{n} \\\ldots & \ldots & \ldots & \ldots \\{\delta_{x}(A)}_{1} & {\delta_{x}(A)}_{2} & \ldots & {\delta_{x}(A)}_{n}\end{bmatrix}_{m \times n}}};$$\left( T_{a}^{b} \right)_{r} = \left\{ {\begin{matrix}{{{{\$ AN\_ CEC}{\_\left\lbrack {r,0} \right\rbrack}} = x_{1}},} \\{{{{\$ AN\_ CEC}{\_\left\lbrack {r,1} \right\rbrack}} = x_{2}},} \\{{\ldots\ldots},} \\{{{{\$ AN\_ CEC}{\_\left\lbrack {r,F} \right\rbrack}} = x_{F}},} \\{{{{\$ AN\_ CEC}{\_ INPUT}{{\_ AXIS}\lbrack r\rbrack}} = a},} \\{{{{\$ AN\_ CEC}{\_ OUTPUT}{{\_ AXIS}\lbrack r\rbrack}} = b},}\end{matrix};} \right.$

Where (T_(C) ^(X))_(h) denotes an r-th T_(C) ^(X) volumetric positioningerror compensation table; m denotes that a moving range of a rotary axisC is divided into m portions, σ_(C) ₁ denotes a first angle, σ_(C) ₂denotes a second angle, and σ_(C) _(m) denotes an m-th angle; h+mdenotes m T_(C) ^(X) volumetric positioning error compensation tables atpresent; (T_(A) ^(X))_(g) denotes a g-th T_(A) ^(X) volumetricpositioning error compensation table; n denotes that a moving range of arotary axis A is divided into n parts; and g+m denotes m T_(A) ^(X)volumetric positioning error compensation tables. (T_(a) ^(b))_(r)denotes an r-th volumetric positioning error compensation table, Fdenotes that an angular range of a reference coordinate axis is dividedinto F equal parts, a denotes the reference coordinate axis (an inputaxis), and b denotes a coordinate axis to be compensated (an outputaxis). If a and b are consistent, then it is linear correlationcompensation. (x₁, x₂, . . . , x_(F)) denotes calculated values ofcoordinates or geometric errors at different positions in F equivalentsor geometric error values, and $AN_CEC_ denotes a special symbol of aCNC system compensation module.

Step (3.3): combined with steps (3.1) and (3.2), the volumetricpositioning error compensation table may be constructed based on allconstituent elements of the linear correlation error terms and thenonlinear correlation error terms, and the volumetric positioning errorcompensation file may be constructed, in which the CNC systemcompensates the entire volumetric positioning errors of the rotary axes.

After constructing the volumetric positioning error compensation filefor the rotary axes of the CNC machine tool, the geometric error valuesof the rotary axes may be filled in sequence. However, a certaindetection error or identification error may be introduced in an actualidentification process, causing a large deviation between an identifiederror compensation value and an actual required error compensationvalue. Therefore, optimization of the geometric error compensation valuemay be performed through a correction manner. For a traditionalcorrection manner of obtaining an average value from a plurality ofinspections and identifications, the efficiency is low while the effectis not necessarily good. In this case, some embodiments of the presentdisclosure take AK4 trajectories recommended by the ISO standard as thelinkage trajectories of the rotary axes of the machine tool, as shown inFIG. 5 , to implement the optimization of the geometrical errorcompensation values in a linkage mode.

Step (4): correction coefficient k and d may be added to the errordatabase obtained in step (2), and correlated with a linkage trajectorypositioning error model to establish a compensation value optimizationmodel for the volumetric positioning errors, the compensation valueoptimization model being configured to optimize the compensation valuesfor the volumetric positioning errors. In some embodiments, thecompensation values for the volumetric positioning errors may beoptimized by the following steps (4.1) to (4.4).

Step (4.1): the correction coefficient k and d may be added to eacherror term in the geometric error vector obtained in step (2), which isrepresented by:

e_Adjust=ke+d;

Where e denotes any error term in the geometric error vector, ande_Adjust denotes a corrected error term.

The significance and the principle of correcting the geometric errorsmay be further illustrated in FIG. 5 . The geometric errors (i.e., theidentified values of the geometric errors) are searched in upper andlower bounds based on the correction coefficients, where k_(w) mainlycompletes scaling adjustment of data, and d_(w) completes overall offsetadjustment of data. The adjustment process may allow approximation ofdetected positioning errors and simultaneous adjustment of all geometricerror terms for accurate error compensation.

An expression of correcting the entire geometric error vectors isrepresented by:

${{Ebase\_ Adjust}_{N \times 12} = \begin{bmatrix}{\overset{\overset{k_{w},d_{w}}{︷}}{\delta_{x}(A)_{1}},\ldots,{\theta_{\gamma}(A)_{1}},{\delta_{x}(C)}_{1},\ldots,\overset{\overset{k_{w},d_{w}}{︷}}{{\theta_{\gamma}(C)}_{1}}} \\\ldots \\{{\delta_{x}(A)}_{N},\ldots,{\theta_{\gamma}(A)_{N}},{\delta_{x}(C)}_{N},\ldots,{\theta_{\gamma}(C)}_{N}}\end{bmatrix}};$

Where Ebase_Adjust denotes a corrected error database, k_(w) and d_(w)denote correction coefficients required for a w-th error term, thepurpose of which is to realize dynamic change of each error term withina certain range, so as to achieve a state closer to a true error.

Step (4.2): the volumetric positioning errors may be calculated in caseof linkage trajectories of rotary axes A and C shown in FIG. 6 based onthe volumetric positioning error model of the tool center point obtainedin step (1), and the volumetric positioning errors may be decomposedinto X, Y, and Z-directions under the workpiece coordinate system (whichis also referred to as the reference coordinate system of the machinetool), so as to obtain the linkage trajectory positioning error model:

${P_{error} = \begin{bmatrix}{P_{{error}\_ X}\left( {\sigma_{A},\sigma_{C},{E(A)},{E(C)}} \right)} \\{P_{{error}\_ Y}\left( {\sigma_{A},\sigma_{C},{E(A)},{E(C)}} \right)} \\{P_{{error}\_ Z}\left( {\sigma_{A},\sigma_{C},{E(A)},{E(C)}} \right)}\end{bmatrix}};$

Where σ_(A), σ_(C) denote the running angles of the rotary axes underthe CNC instruction, E(A) denotes 6 geometric error vectors of a rotaryaxis A, and E(C) denotes 6 geometric error vectors of a rotary axis C.

After taking adding the correction coefficients to the identified valuesof the geometric errors by using the mode of step (4), the correctedgeometric errors Ebase_Adjust may be obtained, which are thensubstituted into the linkage trajectory positioning error model toobtain corrected positioning errors in three directions asP_(error_X_Ad), P_(error_Y_Ad), and P_(error_Z_Ad).

Step (4.3): detected positioning error data (i.e., true positioningerrors) of the linkage trajectories may be obtained by directlydetecting positioning errors of a standard ball center during thelinkage of the rotary axes using the detection instrument (e.g.,R-TEST).

The true positioning error detected in the X-direction of the referencecoordinate system of the machine tool is {circumflex over(P)}_(error_X).

The true positioning error detected in the Y-direction of the referencecoordinate system of the machine tool is {circumflex over(P)}_(error_Y).

The true positioning error detected in the Z-direction of the referencecoordinate system of the machine tool is {circumflex over(P)}_(error_Z).

In the embodiments, an angular interval of the A-axis is 10° and anannular interval of the C-axis is 30°. For the linkage trajectories inFIG. 6 , true positioning errors of 19 detection points may be obtained.

Step (4.4): the true positioning errors obtained by the detectioninstrument may be compared with the corrected positioning errorscalculated by the volumetric positioning error model, as shown in FIG.7A.

A difference between the corrected positioning errors and the truepositioning errors may be calculated, and a minimum of a sum of squaresof the difference may be taken as an optimization objective to obtain 3optimization objectives corresponding to 3 directions of the coordinatesystems. An optimization model for calculating the optimizationobjective is represented by:

$\left\{ \begin{matrix}{F_{x} = \left( {P_{{error}\_ X\_{Ad}} - {\overset{\hat{}}{P}}_{{error}\_ X}} \right)^{2}} \\{F_{y} = \left( {P_{{error}\_ Y\_{Ad}} - {\overset{\hat{}}{P}}_{{error}\_ Y}} \right)^{2}} \\{F_{z} = \left( {P_{{error}\_ Z\_{Ad}} - {\overset{\hat{}}{P}}_{{error}\_ Z}} \right)^{2}}\end{matrix} \right.$

Where F_(x), F_(y), F_(z) denote optimization objectives ofcorresponding geometric error compensation values in case of linkage ofthe rotary axes, all of which indicate that the corrected positioningerrors should be as close as possible to the true positioning errors, soas to achieve the purpose of optimizing the compensation values for thegeometric errors.

Since data in FIG. 7B are all discrete data points, an accurateanalytical solution may not be obtained in data processing. Anapproximate numerical solution may be obtained only by optimizationsearch within a range of values through an intelligent algorithm. Sincethe 3 optimization objectives are all data comparisons of a sametrajectory, and there is no great error weight in one direction, it maybe necessary to optimize the 3 optimization objectives simultaneously.In this regard, the present disclosure introduces the NSGAII algorithmfor fast optimization search of the correction coefficient k and d.

Step (5): compensation quality control may be performed on thecorrection coefficient vectors K and D composed of the correctioncoefficients k and d based on the NSGAII algorithm to realizeoptimization selection of the correction coefficient vectors, so as tocomplete an iterative optimization of the compensation values for thevolumetric positioning errors.

In some embodiments, the optimization selection of the correctioncoefficient vectors may be achieved by the following steps (5.1) to(5.6).

Step (5.1): determination of a combination of correction coefficients:the correction coefficient vector K and D may be determined based on thecorrection coefficient k and d for each geometric error term, which isrepresented by:

$\left\{ {\begin{matrix}{K = \left\lbrack {k_{1},k_{2},\ldots,k_{12}} \right\rbrack} \\{D = \left\lbrack {d_{1},d_{2},\ldots,d_{12}} \right\rbrack}\end{matrix};} \right.$

Where k₁, k₂, . . . , k₁₂, d₁, d₂, . . . , d₁₂ denote correctioncoefficients corresponding to the 12 geometrical errors (which aresorted according to step (2.2)). In this case, an individual particleposition, i.e., a vector consisting of [K, D], has 24 vector columns.

Step (5.2): positioning data pre-processing: positioning data of thestandard ball center when the linkage trajectories of the rotary axesare forward and backward motions may be collected, and processedthree-way positioning error data may be obtained by performing weightedaverage of the positioning data of the forward and backward motions.

Step (5.3): initialization of NSGAII algorithm: parameters of the NSGAIIalgorithm may be set, wherein the algorithmic parameters may include apopulation size M, a total number of external archives R, a crossoveroperator, a selection operator, a mutation operator, a variable range, afitness threshold (tolerance), and a maximum number of iterations(Iterations).

Step (5.4): iterative optimization may be performed based on the NSGAIIalgorithm, which is specifically shown in steps (5.4.1) to (5.4.6)hereinafter.

Step (5.4.1): an initial gene of an individual may be randomly given bya real number encoding manner: the initial gene of the individual may begenerated by using down+(up-down)*rand( ) within a given range ofcorrection coefficient variables. Where down denotes a lower bound of avariable, up denotes an upper bound of the variable, and rand( ) denotesa random number between 0 and 1.

Step (5.4.2): each individual in a population may be evaluated: eachgenerated individual may be substituted into the optimization model instep (4), and each optimization objective value may be directlydesignated as a fitness value to obtain 3 fitness values for eachindividual.

Step (5.4.3): after a certain sorting of each individual, a non-inferiorPareto solution may be stored in an external archive.

In some embodiments, a non-dominated-sort may be performed on eachindividual, including performing the non-dominated-sort on eachindividual to obtain a non-dominated frontier number for eachindividual, and each layer of non-dominated frontier may include adifferent number of individuals.

In some embodiments, a crowding distance may be measured and sorted foreach individual, including sorting the crowding distances of theindividuals on each layer of non-dominated frontier. A crowding distanceformula for the individuals in sorting may be represented by:

L[u] _(d) =L[u] _(d)+(L[u+1]_(v) −L[u−1]_(v))/(f _(v) ^(max) −f _(v)^(min));

Where L[u]_(d) denotes a crowding distance of a u-th individual,L[u+1]_(v) denotes a v-th objective function value of a (u+1)thindividual, and f_(v) ^(max),f_(v) ^(min) denote a maximum value and aminimum value of the v-th objective value, respectively.

In some embodiments, storing the non-inferior solution to the externalarchive may include screening for Pareto solutions from a first frontieruntil the Pareto solutions are greater than a number Num of the externalarchive, and then screening may be stopped.

Step (5.4.4): when a count of iteration is less than a maximum count ofiterations, updating may be performed according to following operationsbased on the general evolution rules of the genetic algorithm: anoperator may be selected based on a binary tournament, and two genes maybe randomly selected from an original parent population for comparison,the priority being given to a smaller sorting rank, and the prioritybeing given to individuals of a same rank with a greater degree ofcrowding; a crossover operator operation may be performed on thepopulation obtained by step (3) according to a probability; a mutationoperator operation may be performed on the population obtained by step(3) according to a probability; and an updated child population may beobtained.

Step (5.4.5): the updated child population may be combined with a childpopulation to form a new population, and step (3) may be iterated toperform secondary evaluation and sorting (i.e., an elite strategy),i.e., better non-inferior solutions may be retained into a nextgeneration.

Step (5.4.6): the external archive may be updated and one cycle may beadded, and steps (5.4.2) to (5.4.5) may be iterated.

Step (5.5): an optimization threshold may be determined: according tothe parameters set in step (5.3), whether a process of iterativeoptimization reaches an objective function fitness threshold or themaximum number of iterations may be determined.

Step (5.6): result output: individual genes may be selected fromobtained external Pareto non-dominated solutions as values of thecorrection coefficient vectors K and D.

Step (6): the volumetric positioning error compensation file of the CNCsystem may be generated based on the correction coefficient vectors andthe error database to implement compensation for the volumetricpositioning errors. After the process of iterative optimization throughthe NSGAII algorithm, relatively ideal correction coefficient vector Kand D may be obtained. The positioning errors at each position of thelinkage trajectories may be calculated by combining with the identifiedvalues of the geometric errors of the rotary axes, and a difference Demay be made between the positioning errors and the detected positioningerrors, and comparison result may be represented in the following table:

X-direction Y-direction Z-direction Maximum Standard Maximum StandardMaximum Standard value deviation value deviation value deviation Before0.028 0.01 0.03 0.014 0.022 0.0095 optimization After 0.016 0.008 0.0260.011 0.019 0.0093 optimization

According to the comparison results in the above table, for thedifference between the positioning errors, both the maximum value andthe standard deviation are reduced after optimization, which shows anobvious optimization effect.

As shown in FIG. 7B, a further comparison graph between a correctedgeometric error and the detected positioning errors is obtained. Thecorrected geometric error is substituted into the sag error compensationfunction of the CNC system to generate the volumetric positioning errorcompensation file for the CNC system, which may realize the compensationfor the volumetric positioning errors of the rotary axes of thefive-axis CNC machine tool, as represented below:

Step (6.1): combined with the CNC system compensation rules from step(3) and step (5), the volumetric positioning error compensation file forcorrecting the geometric error terms may be generated in the followingformat:

$\left( T_{a}^{b} \right)_{rj} = \left\{ {\begin{matrix}{{{{\$ AN\_ CEC}{\_\left\lbrack {r,0} \right\rbrack}} = {{k_{j} \times e_{1}} + b_{j}}},} \\{{{{\$ AN\_ CEC}{\_\left\lbrack {r,1} \right\rbrack}} = {{k_{j} \times e_{2}} + b_{j}}},} \\{{\ldots\ldots},} \\{{{{\$ AN\_ CEC}{\_\left\lbrack {r,F} \right\rbrack}} = {{k_{j} \times e_{F}} + b_{j}}},} \\{{{{\$ AN\_ CEC}{\_ INPUT}{{\_ AXIS}\lbrack r\rbrack}} = a},} \\{{{{\$ AN\_ CEC}{\_ OUTPUT}{{\_ AXIS}\lbrack r\rbrack}} = b},}\end{matrix};} \right.$

Where (e₁, e₂, . . . , e_(F)) denotes an F error value of any one of the12 geometric errors within the angular range, and k_(j), b_(j) denotevalues of the corresponding correction coefficients.

Step (6.2): combined with a nonlinear correlation compensation formatand a linear correlation compensation format of step (3), as well as thecorrected error database Ebase_Adjust, simultaneous compensation may becompleted, which is implemented on the linear correlation error termsand the nonlinear correlation error terms for the positioning errors ofthe tool center point of the CNC machine tool based on step (6.1).

For the five-axis CNC machine tool, relatively high positioning accuracymay be maintained for a considerable period of time after thecompensation of volumetric positioning errors is completed by steps (1)to (6). However, with the increasing use time of machine tool, thepositioning accuracy of the machine tool may inevitably decline, so itis necessary to perform regular accuracy inspection.

Step (7): the geometric error compensation data may be updated to theerror database, the linkage trajectories of the rotary axes may beperiodically detected, and a linkage trajectory positioning errorthreshold may be set, thereby guaranteeing spatial accuracy of thefive-axis CNC machine tool by iteratively implementing detection,optimization, and compensation.

In some embodiments, iteratively implementing detection, optimization,and compensation in step (7) may be realized by steps (7.1) and (7.2) asfollows.

Step (7.1): the linkage trajectory positioning error threshold may beset, a detection cycle may be set according to actual use, and the CNCmachine tool may be driven to detect the volumetric positioning errorsof the linkage trajectories of the rotary axes; if the volumetricpositioning errors do not exceed the linkage trajectory positioningerror threshold, current geometric error compensation values may becontinuously used, otherwise the operation may be performed according tostep (7.2);

Step (7.2): a latest geometric error compensation value may be taken asa new geometric error database, and steps (2) to (6) may be iterated tore-compensate with the latest geometric error compensation value to forma volumetric positioning error system guaranteeing integrated cyclicdetection, optimization, and compensation.

In some embodiments, the CNC system may determine a detection cyclebased on a future compensation time point by predicting the futurecompensation time point at which error compensation may be required inthe future. For example, the CNC system may determine a time intervalbetween two adjacent future compensation time points as a time intervalof the detection cycle. Further descriptions regarding determining thefuture compensation point may be found in FIG. 9 and relateddescriptions thereof.

FIG. 8 is a schematic diagram illustrating an exemplary application of acompensation evaluation model according to some embodiments of thepresent disclosure.

In some embodiments, a CNC system may perform at least one trialmachining after each error compensation to obtain a trial machinedpiece. The trial machined piece may be scored based on a compensationevaluation model to determine an error compensation effect.

The trial machining refers to machining performed to verify the errorcompensation effect. The trial machined piece is a machined pieceobtained after completing the trial machining.

In some embodiments, the CNC system may perform at least one trialmachining after each error compensation, so as to avoid an impact on asubsequent compensation effect due to problems of detection instruments.

In some embodiments, the CNC system may also perform at least one trialmachining after a preset count of error compensations. In someembodiments, the preset count may be determined based on a prioriknowledge or historical data. By intelligently determining the presetcount, the compensation effect may be ideal, and an impact of multipletrial machining on production may be avoided.

The error compensation effect refers to an effect of performing errorcompensation. In some embodiments, the error compensation effect may bein a form of scoring, such as 0-100 points.

The compensation evaluation model refers to a machine learning model. Insome embodiments, the compensation evaluation model may be a machinelearning model with a customized structure as described below. Thecompensation evaluation model may also be a machine learning model ofother structures, such as a neural network model.

In some embodiments, the compensation evaluation model may include aparameter determination layer 810 and an evaluation layer 870.

In some embodiments, the parameter determination layer 810 may be aconvolutional neural network, and the evaluation layer 870 may be aneural network.

In some embodiments, an input of the parameter determination layer 810may include an image detection sequence 810-1 of the trial machinedpiece, and a tool image sequence 810-2, and an output of the parameterdetermination layer 810 may include parameters of the trial machinedpiece 830 and parameters of tool detection 820.

The image detection sequence 810-1 refers to an image sequenceassociated with the trial machined piece. For example, the imagedetection sequence 810-1 may include images of the trial machined piecetaken at various angles. A subject in the images of the trial machinedpiece refers to the trial machined piece.

In some embodiments, the CNC system may obtain the image detectionsequence 810-1 based on an image detection unit. The image detectionunit refers to a unit for obtaining a high-definition image. Forexample, the image detection unit may include a visible light camera, aninfrared camera, or the like.

The tool image sequence 810-2 refers to an image sequence associatedwith a tool. For example, the tool image sequence 810-2 may include toolimages taken from various angles when the tool is not running and toolimages taken continuously when the tool is running. A subject in thetool images refers to the tool.

In some embodiments, the CNC system may obtain the tool image sequence810-2 based on the image detection unit.

The parameters of the trial machined piece 830 refer to parametersrelated to a state of the trial machined piece. For example, theparameters of trial machined piece 830 may include a size, a size of agroove, a depth of the groove of the trial machined piece, etc.

The parameters of tool detection 820 refer to parameters related to astate of the tool. For example, the parameters of tool inspection 820may include a tool size, a wear condition, etc. The wear condition maybe expressed quantitatively.

The parameters of trial machined piece 830 may be obtained after imagerecognition of the image detection sequence 810-1. The parameters oftool detection 820 may be obtained after image recognition of the toolimage sequence 810-2. Ways to perform image recognition include, but arenot limited to, the machine learning model, etc.

In some embodiments, an input of the evaluation layer 870 may includethe parameters of trial machined piece 830, parameters of a standardmachined piece 840, the parameters of tool detection 820, parameters oftool operation 850, and an ambient temperature 860, and an output of theevaluation layer 870 may be machining evaluation data 880.

The parameters of the standard machined piece 840 refer to parametersrelated to a state of the standard machined piece. The standard machinedpiece is a component that is produced to specifications and has astandard shape. In some embodiments, the parameters of the standardmachined piece 840 may be obtained in a preset manner.

In some embodiments, the parameters of tool operation 850 may include atool speed, an advance speed, a cutting angle, etc. In some embodiments,the CNC system may obtain the parameters of tool operation 850 byquerying a database. The parameters of tool operation parameter may bepre-stored in the database. In some embodiments, the CNC system maydetect the parameters of tool operation by using a related device (e.g.,a rotational speed sensor, a shift sensor, an angle sensor, etc.).

In some embodiments, the CNC system may obtain the ambient temperature960 through a temperature sensing device. The temperature sensing devicemay include a thermocouple temperature sensor, a thermistor temperaturesensor, etc.

The machining evaluation data 880 refers to an evaluation of the effectof the trial machining. In some embodiments, the machining evaluationdata 880 may include a machining problem and a compensation effect scoreduring the trial machining. The machining problem may include adimensional deviation, a groove deviation, etc. The compensation effectscore refers to a score of the effect of error compensation on the trialmachined piece. The higher the compensation effect score, the better theeffect of error compensation on the trial machined piece.

In some embodiments, an output of the parameter determination layer 810may be the input of the evaluation layer 870, and the parameterdetermination layer 810 and the evaluation layer 870 may be obtained byjoint training of first training samples with first labels.

In some embodiments, the first training samples may include a sampleimage inspection sequence, a sample tool image sequence, sample standardmachined piece parameters, sample tool operation parameters, and asample ambient temperature, and the first labels may include samplemachining evaluation data corresponding to the first training samples.

In some embodiments, the first training samples may be obtained based onhistorical data. The machining problem in the first labels may beobtained based on a historical trial machining process, and thecompensation effect score may be determined based on a differencebetween the parameters of the trial machined piece (e.g., the size, thesize of the groove, the depth of the groove, etc.) and the parameters ofthe standard machined piece (e.g., the size, the size of the groove, thedepth of the groove, etc.), and the larger the difference, the lower thecompensation effect score.

An exemplary process of joint training may include the followingoperations. The parameters of the trial machined piece and theparameters of tool detection output from an initial parameterdetermination layer may be obtained by inputting the sample imagedetection sequence and the sample tool image sequence into the initialparameter determination layer; and the machining evaluation data outputfrom an initial evaluation layer may be obtained by inputting theparameters of the trial machined piece and the parameters of tooldetection output from the initial parameter determination layer astraining sample data, and the sample standard machined piece parameters,the sample tool operation parameters, and the sample ambient temperatureinto the initial evaluation layer. A loss function may be constructedbased on the first labels and the machining evaluation data output fromthe initial evaluation layer, and parameters of the initial parameterdetermination layer and parameters of the initial evaluation layer maybe synchronously updated. A trained parameter determination layer and atrained evaluation layer may be obtained by parameter updating.

In some embodiments, the CNC system may determine the compensationeffect score for at least one trial machined piece by scoring the atleast one trial machined piece with the compensation evaluation model;and based on the compensation effect score for the at least one trialmachined piece, determine the error compensation effect by calculating amean value or a variance value.

In some embodiments of the present disclosure, a plurality of times oftrial machining may be performed after each error compensation, and thecompensation effect scores of a plurality of trial machined pieces maybe statistically verified to determine the error compensation effect ofthe CNC system, which effectively excludes the influence of accidentalfactors and unstable factors on the error compensation, therebyrealizing accurate detection of the compensation effect and avoiding theinfluence of possible problems of the detection instruments on the errorcompensation. The image detection sequence of the trial machined pieceand the tool image sequence may be processed through the compensationevaluation model, a law may be found from the image detection sequencesof the plurality of trial machined pieces by using a self-learningability of the machine learning model, and a correlation between themachining evaluation data and the image detection sequence of the trialmachined piece may be obtained, thereby improving accuracy andefficiency of determining the machining evaluation data.

In some embodiments, in response to the error compensation effect notmeeting a preset condition, the CNC system may re-perform the volumetricpositioning error compensations.

In some embodiments, the preset condition may be that the errorcompensation effect is greater than a compensation threshold. Forexample, the preset condition may be that a mean value of thecompensation effect score is greater than a mean threshold, or avariance value of the compensation effect score is less than a variancethreshold, etc. The compensation threshold may be manually determined.

In some embodiments, the CNC system may inspect each item beforere-compensating the volumetric positioning errors. For example, items tobe inspected may include the tool, the detection instrument, and a toolcomponent.

In some embodiments, the CNC system may construct a problem featurevector based on the machining problem, the compensation effect score,the parameters of tool detection, and tool operation data to determine adetection item by vector matching.

In some embodiments, the CNC system may determine, based on the problemfeature vector, a reference feature vector in the vector database thatmeets the preset condition, and determine the reference feature vectorthat meets the preset condition as a correlation feature vector; anddetermine a final detection item from a reference detection itemcorresponding to the correlation feature vector. The vector database mayinclude a plurality of reference feature vectors and correspondingreference detection items. The reference feature vector may beconstructed based on a historical machining problem, a historicalcompensation effect score, a historical tool detection parameter, andhistorical tool operation data. The reference detection itemcorresponding to the reference feature vector may be determined based onhistorical detection data. The preset condition refers to adetermination condition used to determine the correlation featurevector. In some embodiments, the preset condition may include that avector distance is less than a distance threshold, a vector distance isminimum, etc.

In some embodiments, the CNC system may re-perform the volumetricpositioning error compensation when a problem occurs to the detectioninstrument and the detection instrument is adjusted. When an otherproblem occurs (e.g., a problem to the tool) and correspondingadjustment is made, the CNC system may re-perform the trial machining,and determine whether to re-perform the volumetric positioning errorcompensation based on whether a new error compensation effect for thetrial machined piece meets the preset condition.

In some embodiments of the present disclosure, by re-performing thevolumetric positioning error compensation in response to the errorcompensation effect not meeting the preset condition, the machiningaccuracy may be improved in an automated and intelligent manner to avoidan accumulation of errors that affects the machining quality of themachined piece.

In some embodiments, the CNC system may improve the calculation accuracyof error compensation calculation when re-performing the volumetricpositioning error compensation.

In some embodiments, the CNC system may increase a count of divisions ofthe motion strokes when re-performing the volumetric positioning errorcompensation.

In some embodiments of the present disclosure, when re-performing thevolumetric positioning error compensation, the count of divisions of themotion strokes may be increased, so that the calculation process of theerror compensation may be more refined, thereby further improving thecompensation effect.

FIG. 9 is a schematic diagram illustrating an exemplary application of acompensation time point prediction model according to some embodimentsof the present disclosure.

In some embodiments, the CNC system may predict a future compensationtime point 970 at which error compensation may be required in thefuture.

The future compensation time point 970 refers to a predicted time pointat which the volumetric positioning error compensation may bere-performed.

In some embodiments, a CNC system may determine the future compensationtime point 970 based on a compensation time point prediction model 960.

The compensation time point prediction model 960 may be a machinelearning model, such as a deep neural network, a convolutional neuralnetwork, or the like, or any combination thereof.

In some embodiments, an input of the compensation time point predictionmodel 960 may include at least one historical error compensation effect910, a pass rate of a current machined piece 920, an error compensationinterval 930, and a cumulative device running duration 940, and anoutput of the compensation time point prediction model 960 may includethe future compensation time point 970.

The historical error compensation effect 910 refers to an errorcompensation effect obtained after performing error compensation over ahistorical time period. The historical time period refers to a timeperiod prior to a current time (i.e., a time point at which theprediction is performed using the compensation time point predictionmodel). Each historical error compensation effect 910 may correspond toa statistical feature (e.g., a mean value, a variance value, etc.) ofthe compensation effect score of a plurality of machined pieces obtainedover the historical time piece. One or more historical time intervalsmay be included in the historical time period. A time length of eachhistorical time period and each historical time interval may be either asystem default or a manual preset value.

The pass rate of the current machined piece 920 refers to a pass rate ofa plurality of machined pieces after error compensation according to asame geometric error compensation value. In some embodiments, the passrate of the current machined piece 920 may be determined based on a passcriterion. For example, a ratio of a count of the machined pieces in theplurality of machined pieces that meet the pass criterion to a totalcount of machined pieces may be determined as the pass rate. The passcriterion may be that a dimension of the machined piece and a size or adepth of a groove meet requirements of the standard machined piece andare within tolerance.

The error compensation interval 930 refers to a time interval between alast error compensation and a current error compensation. The CNC systemmay determine the error compensation interval 930 based on a time pointof the last error compensation and the current time point.

The cumulative device running duration 940 refers to a duration of thefive-axis CNC machine tool from a start time of operation to the currenttime point. The CNC system may determine the cumulative device runningduration 940 based on the start time of operation of the five-axis CNCmachine tool and the current time point.

In some embodiments, an input of the compensation time point predictionmodel 960 may further include a historical error compensation interval950.

The historical error compensation interval 950 refers to a time intervalbetween two adjacent error compensations in a historical time period.The CNC system may determine the historical error compensation interval950 based on historical data. Further description regarding thehistorical time period may be found hereinabove.

Considering that the shorter the error compensation interval, the closerthe future compensation time point to the current time point. Byinputting the historical error compensation interval as an output of thecompensation time point prediction model, a determined futurecompensation time point may be more comprehensive and accurate.

In some embodiments, the compensation time point prediction model 960may be obtained by training of second training samples with secondlabels. The second training samples may include a sample historicalerror compensation effect before a sample time point, a pass rate of asample machined piece, a sample error compensation interval, and asample cumulative device running duration. The second labels may be asample compensation time point after the sample time point. The secondtraining samples may be obtained based on the historical data. Thesecond label may be obtained based on historical detection data. Forexample, the CNC machine tool may be driven daily to detect thevolumetric positioning errors until the volumetric positioning errors ata time point exceed the linkage trajectory positioning error threshold,and the time point may be recorded as the second label.

When the input of the compensation time point prediction model alsoincludes the historical error compensation interval, the second trainingsamples may also include a sample historical error compensationinterval.

An exemplary training process may include following operations. Aplurality of second training samples with second labels may be inputinto an initial compensation time point prediction model, a lossfunction may be constructed from the second labels and results of theinitial compensation time point prediction model, and parameters of theinitial compensation time point prediction model may be iterativelyupdated based on the loss function. When the loss function of theinitial compensation time point prediction model meets a presetcondition, model training may be completed, and a trained compensationtime point prediction model may be obtained. The preset condition may bethat the loss function converges, a count of iterations reaches athreshold, etc.

If the detection cycle is too long, it may lead to too much erroraccumulation, affecting the machining accuracy. But if the detectioncycle is too short, it may make a detection and compensation processcumbersome, causing a waste of time. In some embodiments of the presentdisclosure, by predicting the future compensation time point at whicherror compensation is required in the future, errors and waste caused bya fixed detection cycle may be avoided, and accuracy of errorcompensation may be improved to ensure the machining accuracy and tosave time and resources. At least one historical error compensationeffect, the pass rate of the current machined piece, the errorcompensation interval, and the cumulative device running duration may beprocessed through the compensation time point prediction model, a lawmay be found from a plurality of historical error compensation effectsby using a self-learning ability of the machine learning model, and acorrelation between the future compensation time point and thehistorical error compensation effect may be obtained, thereby improvingthe accuracy and efficiency of determining the future compensation timepoint.

Having thus described the basic concepts, it may be rather apparent tothose skilled in the art after reading this detailed disclosure that theforegoing detailed disclosure is intended to be presented by way ofexample only and is not limiting. Although not explicitly stated here,those skilled in the art may make various modifications, improvements,and amendments to the present disclosure. These alterations,improvements, and modifications are intended to be suggested by thisdisclosure and are within the spirit and scope of the exemplaryembodiments of this disclosure.

Moreover, certain terminology has been used to describe embodiments ofthe present disclosure. For example, the terms “one embodiment,” “anembodiment,” and/or “some embodiments” mean that a particular feature,structure, or feature described in connection with the embodiment isincluded in at least one embodiment of the present disclosure.Therefore, it is emphasized and should be appreciated that two or morereferences to “an embodiment” or “one embodiment” or “an alternativeembodiment” in various portions of the present disclosure are notnecessarily all referring to the same embodiment. In addition, somefeatures, structures, or characteristics of one or more embodiments inthe present disclosure may be properly combined.

Furthermore, the recited order of processing elements or sequences, orthe use of numbers, letters, or other designations therefore, is notintended to limit the claimed processes and methods to any order exceptas may be specified in the claims. Although the above disclosurediscusses some embodiments of the invention currently considered usefulby various examples, it should be understood that such details are forillustrative purposes only, and the additional claims are not limited tothe disclosed embodiments. Instead, the claims are intended to cover allcombinations of corrections and equivalents consistent with thesubstance and scope of the embodiments of the invention. For example,although the implementation of various components described above may beembodied in a hardware device, it may also be implemented as a softwareonly solution, e.g., an installation on an existing server or mobiledevice.

Similarly, it should be appreciated that in the foregoing description ofembodiments of the present disclosure, various features are sometimesgrouped together in a single embodiment, figure, or description thereoffor the purpose of streamlining the disclosure aiding in theunderstanding of one or more of the various embodiments. However, thisdisclosure does not mean that object of the present disclosure requiresmore features than the features mentioned in the claims. Rather, claimedsubject matter may lie in less than all features of a single foregoingdisclosed embodiment.

In some embodiments, the numbers expressing quantities or propertiesused to describe and claim certain embodiments of the present disclosureare to be understood as being modified in some instances by the term“about,” “approximate,” or “substantially.” For example, “about,”“approximate” or “substantially” may indicate ±20% variation of thevalue it describes, unless otherwise stated. Accordingly, in someembodiments, the numerical parameters set forth in the writtendescription and attached claims are approximations that may varydepending upon the desired properties sought to be obtained by aparticular embodiment. In some embodiments, the numerical parametersshould be construed in light of the number of reported significantdigits and by applying ordinary rounding techniques. Notwithstandingthat the numerical ranges and parameters setting forth the broad scopeof some embodiments of the present disclosure are approximations, thenumerical values set forth in the specific examples are reported asprecisely as practicable.

Each of the patents, patent applications, publications of patentapplications, and other material, such as articles, books,specifications, publications, documents, things, and/or the like,referenced herein is hereby incorporated herein by this reference in itsentirety for all purposes. History application documents that areinconsistent or conflictive with the contents of the present disclosureare excluded, as well as documents (currently or subsequently appendedto the present specification) limiting the broadest scope of the claimsof the present disclosure. By way of example, should there be anyinconsistency or conflict between the description, definition, and/orthe use of a term associated with any of the incorporated material andthat associated with the present document, the description, definition,and/or the use of the term in the present document shall prevail.

In closing, it is to be understood that the embodiments of the presentdisclosure disclosed herein are illustrative of the principles of theembodiments of the present disclosure. Other modifications that may beemployed may be within the scope of the present disclosure. Thus, by wayof example, but not of limitation, alternative configurations of theembodiments of the present disclosure may be utilized in accordance withthe teachings herein. Accordingly, embodiments of the present disclosureare not limited to that precisely as shown and described.

1. A method of fast optimization and compensation for volumetricpositioning errors of rotary axes of a five-axis computer numericalcontrol (CNC) machine tool, comprising: (1) establishing a volumetricpositioning error model for the rotary axes of the five-axis CNC machinetool based on geometric errors of the rotary axes; (2) completing anidentification of the 12 geometric errors of the rotary axes based oninstrument detection data, and forming an error database containing the12 geometric error vectors; obtaining a geometric error identificationresult by performing the identification of the geometric errors using adetection instrument and a relevant identification algorithm based onthe volumetric positioning error model established for the rotary axesof the five-axis CNC machine tool based on step (1); wherein each of the12 geometric error vectors is an error vector combined by 6 geometricerrors identified for each rotary axis:$E_{1 \times 12} = \left\lbrack {\underset{\underset{6}{︸}}{{\delta_{x}(A)},\ldots,{\theta_{\gamma}(A)}},\underset{\underset{6}{︸}}{{\delta_{x}(C)},\ldots,{\theta_{\gamma}(C)}}} \right\rbrack$the geometric errors are related to a motion position, and each of thegeometric errors is turned into a vector related to the motion position,i.e., when a motion stroke of each rotary axis is divided into N equalparts, a same error element is slightly different at different rotationangle positions, which is noted as:${{{Ebas}e_{N \times 12}} = \begin{bmatrix}{{\delta_{x}(A)_{1}},\ldots,{\theta_{\gamma}(A)}_{1},{\delta_{x}(C)}_{1},\ldots,{\theta_{\gamma}(C)}_{1}} \\\ldots \\{{\delta_{x}(A)_{N}},\ldots,{\theta_{\gamma}(A)}_{N},{\delta_{x}(C)}_{N},\ldots,{\theta_{\gamma}(C)}_{N}}\end{bmatrix}};$ where Ebase_(N×12) denotes a matrix of N rows and 12columns, which is a geometric error database of the five-axis CNCmachine tool; (3) decomposing the volumetric positioning errors of therotary axes into linear correlation and nonlinear correlation, andconstructing a volumetric positioning error compensation table of therotary axes of a CNC system by combining with a sag error compensationfunction of the CNC system; (4) adding correction coefficients k and dto the geometric error database obtained in step (2) and correlating thecorrection coefficients k and d with the volumetric positioning errormodel for the rotary axes to establish a compensation data optimizationmodel for the volumetric positioning errors of the five-axis CNC machinetool; (5) carrying out compensation quality control on correctioncoefficient vectors K and D composed of the correction coefficients kand d based on a non-dominated sorting genetic algorithm (NSGAIIalgorithm) so as to complete an iterative optimization of thecompensation data to realize an optimization of the correctioncoefficients; (6) generating a volumetric positioning error compensationfile for the CNC system by using the correction coefficient vectors andthe error database containing 12 geometric errors obtained in step (5),so as to complete the compensation of the geometric errors of the rotaryaxis of the five-axis CNC machine tool; and (7) updating geometric errorcorrection data to the error database, and periodically detectinglinkage trajectories of the rotary axes, and setting a linkagetrajectory positioning error threshold, thereby guaranteeing spatialaccuracy of the five-axis CNC machine tool by iteratively implementingdetection, optimization, and compensation.
 2. The method of claim 1,wherein the process of establishing the volumetric positioning errormodel of a tool center point caused by the geometric errors of therotary axes of the five-axis CNC machine tool in step (1) includes: forany type of the rotary axis, due to an effect of the geometric error ofthe rotary axis, an actual motion matrix of the rotary axis is:T _(i) =R(i)·E(i),i=A,B,C where T_(i) denotes an actual motiontransformation matrix of i axis, R(i) denotes a theoretical motiontransformation matrix of the i axis, and E(i) denotes a geometric errormatrix, which are specifically described as follows:${{R(A)} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos A} & {{- s}{in}A} & 0 \\0 & {\sin A} & {\cos A} & 0 \\0 & 0 & 0 & 1\end{bmatrix}},{{R(B)} = \begin{bmatrix}{\cos B} & 0 & {\sin B} & 0 \\0 & 1 & 0 & 0 \\{{- s}{in}B} & 0 & {\cos B} & 0 \\0 & 0 & 0 & 1\end{bmatrix}},$ ${{R(C)} = \begin{bmatrix}{\cos C} & {{- s}{in}C} & 0 & 0 \\{\sin C} & {\cos C} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}},$ ${E_{i} = \begin{bmatrix}1 & {- {\theta_{\gamma}(i)}} & {\theta_{\beta}(i)} & {\delta_{x}(i)} \\{\theta_{\gamma}(i)} & 1 & {\theta_{\alpha}(i)} & {\delta_{y}(i)} \\{- {\theta_{\beta}(i)}} & {\theta_{\alpha}(i)} & 1 & {\delta_{z}(i)} \\0 & 0 & 0 & 1\end{bmatrix}},{i = A},B,{C;}$ where A, B, C denote a rotating angle ina CNC instruction, (δ_(x)(i), δ_(y)(i), δ_(z)(i)) respectively denote adisplacement error along X, Y, and Z-directions, and (θ_(α)(i),θ_(β)(i), θ_(γ)(i)) respectively denote angular errors around the X, Y,and Z-axes; since the five-axis CNC machine tool has only two rotaryaxes involved in the motion, a kinematic relationship of the tool centerpoint in the presence of errors in linkage operation of the C-axis andthe A-axis as follows:P _(actual) =T(C)·T(A)·P _(t); where P_(actual) denotes actualcoordinates of the tool center point under a workpiece coordinatesystem, P_(t)=[0 0 −L 1]^(T); ignoring the geometric error matrix forthe above equation, which is a theoretical motion mode, the kinematicrelationship of the tool center point in an ideal case is obtained as:P _(ideal) =R(C)·R(A)·P _(t); where P_(ideal) is theoretical coordinatesof the tool center point under the workpiece coordinate system; thevolumetric positioning error model for the rotary axis is furtherrepresented by:P _(error) =P _(actual) −P _(ideal); by substituting the motion matrixand the geometric error matrix E_(i), the volumetric positioning errormodel is further represented by: $P_{error} = \begin{bmatrix}{{{\delta_{x}(A)}\cos\sigma_{C}} - {{\delta_{y}(A)}\sin\sigma_{C}} - {L{\theta_{\beta}(A)}\cos\sigma_{A}}} \\{{\cos\sigma_{C}} - {L\theta_{\alpha}(A)\cos\sigma_{A}\sin\sigma_{C}} -} \\{{L{\theta_{\gamma}(C)}\sin\sigma_{A}\cos\sigma_{C}} - {L{\theta_{\beta}(C)}\cos\sigma_{A}} -} \\{{L{\theta_{\gamma}(A)}\sin\sigma_{A}\cos\sigma_{C}} + {\delta_{x}(C)}} \\{{\delta_{\gamma}(C)} + {{\delta_{x}(A)}\sin\sigma_{C}} + {L{\theta_{\alpha}(C)}\cos\sigma_{A}} -} \\{{L{\theta_{\gamma}(A)}\sin\sigma_{A}\sin\sigma_{C}} + {{\delta_{\gamma}(A)}\cos\sigma_{C}} +} \\{{L{\theta_{\alpha}(A)}\cos\sigma_{A}\cos\sigma_{C}} - {L{\theta_{\gamma}(C)}\sin\sigma_{A}\sin\sigma_{C}} -} \\{L{\theta_{\beta}(A)}\cos\sigma_{A}\cos\sigma_{C}} \\{{L{\theta_{\beta}(A)}\sin\sigma_{A}\sin\sigma_{C}} + {L{\theta_{\alpha}(C)}\sin\sigma_{A}\cos\sigma_{C}} +} \\{{L\theta_{\alpha}(A)\sin\sigma_{A}} + {\delta_{z}(C)} + {\delta_{z}(A)}} \\0\end{bmatrix}$
 3. The method of claim 2, wherein the detectioninstrument in the step (2.1) is an R-TEST or ball-bar.
 4. The method ofclaim 3, wherein the process of constructing the volumetric positioningerror compensation table of the CNC system in step (3) includes: (3.1)dividing the volumetric positioning errors into the linear correlationand the nonlinear correlation by analyzing a mathematical model betweenthe volumetric positioning errors of the tool center point and thegeometric errors of the rotary axes; (3.2) realizing compensation ofnonlinear correlation error terms based on the sag error compensationfunction of the CNC system; (3.3) combining steps (3.1) and (3.2),establishing the volumetric positioning error compensation table basedon all constituent elements of the linear correlation error terms andthe nonlinear correlation error terms, and constructing the volumetricpositioning error compensation file for the CNC system to compensate theentire volumetric positioning errors of the rotary axes.
 5. The methodof claim 4, wherein the process of establishing the compensation dataoptimization model for the volumetric positioning errors of the CNCmachine tool in step (4) includes: (4.1) adding the correctioncoefficients k and d to each error term of the geometric error vectorsobtained in step (2); (4.2) decomposing the equation of the volumetricpositioning error of the tool center point obtained in step (1), whichis the volumetric positioning errors of linkage of the rotary axes A andC in the five-axis CNC machine tool, in the X, Y and Z directions undera reference coordinate system of a machine tool; obtaining correctedgeometric errors by adding the correction coefficients to the identifiedgeometric errors by means of step (4), and then obtaining correctedpositioning errors in the three directions by substituting the correctedgeometric errors into the linkage trajectory positioning errors of therotary axes; (4.3) obtaining true positioning error data of the machinetool by directly detecting a positioning error of a standard ball centerduring the linkage of the rotary axes using the detection instrument;and (4.4) calculating a difference between the corrected positioningerrors in step (4.2) and the true positioning errors in step (4.3), andtaking the smallest sum of squares of the difference as an optimizationobjective to obtain 3 optimization objectives.
 6. The method of claim 5,wherein the step (5) includes: (5.1) a combination of correctioncoefficients: determining the correction coefficient vectors K and D byadding the correction coefficients k and d for each geometric errorterm; (5.2) positioning data pre-processing: collecting positioning dataof the standard ball center when the linkage trajectories of the rotaryaxes are forward and backward motions, and obtaining processed three-waypositioning error data by performing weighted average of the positioningdata of the forward and backward motions; (5.3) initialization of NSGAIIalgorithm and setting of parameters of the NSGAII algorithm: thealgorithmic parameters include a population size M, a total number ofexternal archives R, a crossover operator, a selection operator, amutation operator, a variable range, a fitness threshold (tolerance),and a maximum number of iterations (Iterations); (5.4) iterativeoptimization of the NSGAII algorithm; (5.5) determining an optimizationthreshold: according to the parameters set in step (5.2), determiningwhether a process of iterative optimization reaches an objectivefunction fitness threshold or the maximum number of iterations; and(5.6) result output: selecting individual genes from obtained externalPareto non-dominated solutions as values of the correction coefficientvectors K and D according to a principle.
 7. The method of claim 6,wherein the process of performing iterative optimization of the NSGAIIalgorithm in step (5.4) includes: (5.4.1) selecting a real number codingmethod to randomly give initial genes of each individual: generating theinitial genes of an individual by using down+(up-down)*rand( ) withinthe range of a given correction coefficient variable; wherein downrepresents a lower bound of the variable, up represents an upper boundof the variable, and rand( ) is a random number between 0 and 1; (5.4.2)evaluating each individual in the population: substituting eachgenerated individual into the optimization model in step (4), anddirectly using each optimization objective value as a fitness value toobtain 3 fitness values corresponding to each individual; (5.4.3)storing non-inferior Pareto solutions in the external archives aftersorting each individual; (5.4.4) using generalized evolutionary rules ofthe genetic algorithm when a count of cycles is less than a maximumcount of cycles; (5.4.5) combining the updated offspring population withan original parent population to form a new population, and repeatingstep (5.4.3) to perform secondary evaluation and sorting, i.e.,retaining better non-inferior solutions to the next generation; and(5.4.6) updating the external archives and adding one cycle, anditerating the step (5.4.2) to the step (5.4.5).
 8. The method of claim7, wherein the process of compensating for volumetric positioning errorsof a five-axis CNC machine tool in step (6) includes: (6.1) generatingthe volumetric positioning error compensation file of the correctedgeometric errors by combining the contents of steps (3) and (5); and(6.2) accomplishing simultaneous compensation of the linear correlationerror terms and the nonlinear correlation error terms of the positioningerrors of the tool center point of the five-axis CNC machine tool basedon the step (6.1) according to the corrected error database of therotary axes.
 9. The method of claim 8, wherein a process of iterativelyimplementing detection, optimization, and compensation in step (7)includes: (7.1) setting the linkage trajectory positioning errorthreshold, setting a detection cycle duration according to an actualuse, and driving the CNC machine tool to detect the linkage trajectorypositioning error; if the linkage trajectory positioning error thresholdis not exceeded, continuing to use, otherwise performing according tostep (7.2); and (7.2) taking a latest geometric error compensation dataas a new error database, and iterating steps (2) to (6) to re-compensatewith the latest geometric error compensation data to form a volumetricpositioning error system guaranteeing integrated cyclic detection,optimization, and compensation.